In your method or any lifecycle hook that changes the array item at particuar index

new Vue({
    el: '#app',
    data:{
        myArr : ['apple', 'orange', 'banana', 'grapes']
    },
    methods:{
        changeArrayItem: function(){
            //this will not work
            //myArr[2] = 'strawberry';
            
            //Vue.$set(array, index, newValue)
            this.$set(this.myArr, 2, 'strawberry');
        }
    }
}) 

Here is the link to the fiddle

You can perform the same change instead of using Vue.$set by using the Array prototype's splice()

new Vue({
    el: '#app',
    data:{
        myArr : ['apple', 'orange', 'banana', 'grapes']
    },
    methods:{
        changeArrayItem: function(){
            //this will not work
            //myArr[2] = 'strawberry';
            
            //Array.splice(index, 1, newValue)
            this.myArr.splice(2, 1, 'strawberry');
        }
    }
}) 

If yoi have nested array, the following can be done

new Vue({
    el: '#app',
    data:{
        myArr : [
            ['apple', 'banana'],
            ['grapes', 'orange']
        ]
    },
    methods:{
        changeArrayItem: function(){
            this.$set(this.myArr[1], 1, 'strawberry');
        }
    }
})

Here is the link to the jsfiddle

new Vue({
    el: '#app',
    data:{
        myArr : [
            {
                name: 'object-1',
                nestedArr: ['apple', 'banana']
            },
            {
                name: 'object-2',
                nestedArr: ['grapes', 'orange']
            }
        ]
    },
    methods:{
        changeArrayItem: function(){
            this.$set(this.myArr[1].nestedArr, 1, 'strawberry');
        }
    }
}) 

Here is the link to the fiddle